The Simulation Optimization approach is applicable to very general multi-location inventory systems. The concept presented in this chapter iteratively combines a sim- ulator with Particle Swarm Optimization. This concept allows the investigation of complex models with few assumptions and is theoretically not limited to a loca- tion count contrary to analytical approaches. Due to the complexity of the model, it is difficult to understand the effect of certain policies. Therefore, valuable insights regarding the dynamics of the system are obtained through simulation in addition to the optimal parameter set. However, applying global optimization to complex models still involves a certain risk to end in a local optimum. That risk is confined by extending the simulation time and the optimization cycle count. The optimum
depends on the model specification and shows a specific structure. The development of such a structure is one of the most intriguing aspects, and the question arises, what conditions have a promoting effect.
As aforementioned an advantage of the Simulation Optimization of multi-location inventory systems with lateral transshipments is that the model itself is straightfor- ward extendable. Functional extensions are, e.g., policies for periodic orders, trans- shipment orders and product offers. Extending the parameter set itself, the capacity of the locations can be optimized by introducing estate and energy cost for unused storage. Thus, not only the flows of transshipments are optimized, but also the al- location of capacities. In addition to static aspects of the model, the parameter set may be extended by dynamic properties such as the location-specific order period time. Besides these extensions there is an idea regarding orders from more than one location at a time. Under specific circumstances one location evolves as a supplier, ordering and redistributing product units. Therefore, the basic idea is to release an order by several locations and to solve the Traveling Salesman Problem with mini- mal cost. However, the existing heuristics already seem to approximate such a trans- portation logic well, and thus, the inclusion of more elaborate policies is expected just to increase complexity. Further research may also concentrate on characteristics favoring demand forecast and promoting certain flows through a location network leading to a structure.
Acknowledgment.The authors would like to thank the Robert Bosch doctoral program, the German Academic Exchange Service and the Foundation of the German Business for fund- ing their research.
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Traditional and Hybrid Derivative-Free Optimization Approaches for Black Box Functions
Genetha Anne Gray and Kathleen R. Fowler
Abstract.Picking a suitable optimization solver for any optimization problem is quite challenging and has been the subject of many studies and much debate. This is due in part to each solver having its own inherent strengths and weaknesses. For example, one approach may be global but have slow local convergence properties, while another may have fast local convergence but is unable to globally search the entire feasible region. In order to take advantage of the benefits of more than one solver and to overcome any shortcomings, two or more methods may be combined, forming a hybrid. Hybrid optimization is a popular approach in the combinatorial optimization community, where metaheuristics (such as genetic algorithms, tabu search, ant colony, variable neighborhood search, etc.) are combined to improve robustness and blend the distinct strengths of different approaches. More recently, metaheuristics have been combined with deterministic methods to form hybrids that simultaneously perform global and local searches. In this Chapter, we will exam- ine the hybridization of derivative-free methods to address black box, simulation- based optimization problems. In these applications, the optimization is guided solely by function values (i.e.not by derivative information), and the function values re- quire the output of a computational model. Specifically, we will focus on improving derivative-free sampling methods through hybridization. We will review derivative- free optimization methods, discuss possible hybrids, describe intelligent hybrid ap- proaches that properly utilize both methods, and give an examples of the successful application of hybrid optimization to a problem from the hydrological sciences.
Genetha Anne Gray
Department of Quantitative Modeling & Analysis, Sandia National Laboratories,
P.O. Box 969, MS 9159, Livermore, CA 94551-0969 USA e-mail:gagray@sandia.gov
Kathleen R. Fowler
Department of Mathematics & Computer Science,
Clarkson University, P.O. Box 5815, Potsdam, NY, 13699-5815 USA e-mail:kfowler@clarkson.edu